lesson
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If you have a square plot of land that is 7 feet on each side, its area is 72=49 square feet. But what happens if we break that 7 into 3+4?
Is (3+4)2 just 32+42? If we calculate 9+16, we only get 25 — where did the other 24 square feet go?
Visualizing a Square with Side (a+b)
The area of any rectangle or square is found by multiplying its length by its width.
When we build a large square with side length (a+b), we can split each side into two segments of lengths a and b.
📊Create an interactive visual diagram of an area model for (a+b)^2. Show a large square divided into 4 colored quadrants: top-left square with side 'a' (colored soft blue, labeled area 'a²'), top-right rectangle with sides 'b' and 'a' (colored soft orange, labeled area 'ab'), bottom-left rectangle with sides 'a' and 'b' (colored soft orange, labeled area 'ab'), and bottom-right square with side 'b' (colored soft green, labeled area 'b²'). Dimension labels 'a' and 'b' are clearly marked on top, bottom, left, and right borders with bracket indicators. Include a summary box below that highlights the sum: Total Area = a² + ab + ab + b² = a² + 2ab + b².
How do these four separate puzzle pieces combine into a single algebraic formula?
Deriving the Identity
The total area of the large square is (a+b)×(a+b), which we write as (a+b)2.
Adding up the four inner regions gives one square of area a2, two identical rectangles of area ab, and one small square of area b2.
(a+b)2=a2+ab+ab+b2=a2+2ab+b2
Around 300 BCE, the Greek mathematician Euclid proved this exact geometric relationship in Proposition 4 of Elements, Book II, centuries before modern algebraic symbols existed.
How do we use this identity to expand expressions with variables and numbers?